Advanced Problem Archive

* Denotes a problem which has not yet been solved.
A problem whose solution has recently been posted is indicated with a +.
  1. A Bitangent to a Quartic Curve
  2. Minimizing the Area of a Parabolic Sector
  3. Finding the Height of a Mountain
  4. A Circle Bitangent to a Parabola
  5. The Area and Perimeter of a Certain Region
  6. An Incorrect Quotient Rule for Derivatives
  7. Which Value of a Gives a Unique Solution?
  8. Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres)
  9. Expected Area of an Inscribed Triangle
  10. The Limit of a Harmonic Sum
  11. Maximize the Area Between a Circle and a Polygon
  12. Maximize the Total Surface Area of an Inscribed Cylinder and Cone
  13. Two Infinite Series
  14. The Number of Solutions of a Trigonometric Equation
  15. Drilling a Cube
  16. Overlapping Tangent Parabolas
  17. Surface Area and Volume of a Pendant
  18. Minimize the Arclength
  19. When Does the Sum Get Big?
  20. Two Leibnizian Infinite Series
  21. A Product of Cosines Equation
  22. Minimizing the Perimeter of a Parabolic Region
  23. Half Full or Half Empty?
  24. A Putnam Problem
  25. Intersecting Cylinders, Part I
  26. Intersecting Cylinders, Part II
  27. A Heart-Shaped Box
  28. Probability of Making a Triangle
  29. Maximize the People-Days
  30. Making a Planter
  31. What's the Average Width?
  32. Turning the Corner with a Rod
  33. Putting a Sphere in a Cone
  34. Average Size of a Piece of a Broken Stick
  35. Rotating Reuleaux
  36. Find the Limit of the Sum (Putnam)
  37. The Average Distance Between Points in the Cantor Set
  38. Put the Heart in a Box
  39. Where Does the Spiral End?
  40. What Shape Does the Sliding Rod Trace Out?
  41. The Expected Area of a Triangle
  42. Another Limit Problem
  43. Find the Area of the Deltoid
  44. Two Cyclists
  45. Three Circles in a Parabola
  46. A Circle Perpendicular to a Parabola
  47. The Smallest Triangle Containing Two Unit Circles
  48. Area of a Lissajous Figure
  49. Another Polygonal Spiral
  50. What's the Area of the Bishop's Hat?
  51. What's the Probability that Alice Makes Fewer Tosses than Bob?
  52. Sum the Series
  53. Buffon's Bent Needle
  54. A Weight on a Rope
  55. Spreading Rumors
  56. Yet Another Infinite Series
  57. The Smallest Circle Containing Two Ellipses
  58. A Tetrahelix (from H.S.M. Coxeter)
  59. The Area Common to Two Roses
  60. Find the Parabola Given Three Tangent Lines
  61. Again with the Infinite Series?
  62. Area Equals Arclength
  63. Buffon's Dartboard
  64. Light Reflecting in an Ellipse
  65. Maximize the Tetrahedron's Volume
  66. Compute the Area of a Triangle from the Area of its Shadows
  67. An Infinite Series involving the Inverse Tangent
  68. Find the Volume of the Intersecting Cones
  69. A Differential Equation
  70. Find the Volume and Surface Area of this Solid
  71. A Tribute to Martin Gardner
  72. Find the Determinant of the Matrix
  73. How Much Green is There in the Wachovia Logo?
  74. The Locus of Certain Bitangent Points
  75. How Tall Can the Stack of Hemispheres Be?
  76. Evaluate the Integral
  77. An Infinite Series with Factorials
  78. Find the Ellipse with Least Area Containing Two Unit Circles, etc.
  79. A Solid with a Square Shadow, a Circular Shadow, and a Triangular Shadow
  80. What's the Area the Particle Can Reach?
  81. What's the Probability of Forming a Triangle? *
  82. Find the Volume of the Great Rhombicuboctahedron
  83. A Limit Involving Partial Sums of the Exponential Function
  84. A Generalization of the Hyperbolic Trig Functions
  85. A Trigonometric Problem
  86. Find the Polynomial
  87. Find the Plane Tangent to the Three Paraboloids +
  88. Find the Area of the Hexagon Formed by Side-Trisectors
  89. If You Divide a Cube into Eight Pieces with Three Perpendicular Planes, How Big Can the Largest Piece Be? *
  90. Integrating Powers of the Cantor Function
  91. Three Mutually Perpendicular Lines That Pass Through a Circle
  92. A Problem from Group Theory *
  93. Minimize the Sum of the Reciprocals of the Distances Between Four Points on a Sphere
    * This problem has been completely reformulated.
  94. Two Infinite Series +
  95. Find the Volume of a Region Near a Cone
  96. What's the Expected Length of a Slice of a Rectangle?
  97. How Far (on Average) to the Circle's Rim?
  98. An Integral Involving the Fractional Part Function
  99. Find the Area Inside the Curve
  100. Two Infinite Chains of Circles
  101. A Limit Involving Two Trigonometric Integrals
  102. Minimize the Product of the Lengths of Two Perendicular Diameters of an Ellipse
  103. A Limit Involving the Solution of a Trigonometric Equation
  104. A Circle Rolling Down a Ramp
  105. Find the Area Inside the Parametrized Curve
  106. Find the Volume Inside the Two Ellipsoids
  107. What's the Expected Distance Between Two Points on a Sphere? (Putnam)
  108. Happy New Year, 2008 +
  109. A Nested Radical Question
  110. Find the Limit of the Recursive Sequence
  111. Minimize the Length of the Normal to a Paraboloid
  112. Evaluate the Infinite Divisorial Expression

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